FP2 Flashcards
What is sin⍺cosβ equal to
0.5 (sin(⍺+β) + sin(⍺-β))
What is sin(⍺)sin(β) equal to?
0.5(cos(⍺-β)-cos(⍺+β))
What is cos(⍺)cos(β) equal to?
0.5(cos(⍺-β)+cos(⍺+β))
in what form do you put cos²x for integration?
(cos 2x +1) /2
in what form do you put sin²x for integration?
(1-cos2x)/2
what is arccot(x) equal to?
arctan(1/x).
true for other trig functions.
What is x equal to when converting to polar form?
r*cos theta
what is sinθ equal to when converting to Cartesian form?
y/r
what is x² +y² equal to when converting to polar form
r²
what is tanθ equal to when converting to polar form
y/x
how is the area of a polar graph found?
intergrate 0.5 r²
What does r=a*secθ represent?
a straight vertical line
what does r=a*cosecθ represent
a straight horizontal line
what is the general solution to sinx=y
x=nπ+(-1)^n*arcsin(y)
what is the general solution to cosx=y
x=2nπ±arcos(y)
what is the general solution to tan(x)=y
x=nπ+arctan(y)
what is the principle polar co-ordinate ?
r is greater than 0 and -π
What is tanh(x) equal to?
Sinh(x)/cosh(x)
What is coth x
1/ tanh x
What is sech x
1/cosh x
What is cosech x
1/sinh x
If you have the matrices A and B and the in verses of each, what is (AB)⁻¹ equal to?
(AB)⁻¹ = B⁻¹ * A⁻¹
What are the 2 identities used to lower the powers of hyperbolic functions?
2sinh²(x) = cosh(2x)-1 2cosh²(x)= cosh(2x)+1
What is the trace of a matrix and how does this relate to eigenvectors?
The trace is the sum of the leading diagonal and this equals the sum of the eigenvectors.
How do you express a+bi in the form re^i⊝?
r is |a+bi|
⊝= arg(a+bi)
What is polar form?
Z=r(cos⊝+isin⊝)
How do you multiply numbers in polar form?
r of the product is the r₁×r₂
⊝ of the product = ⊝₁+⊝₂
What is de moivre’s theorem?
(Cos⊝+isin⊝)ª= cos a⊝ +isin a⊝
What is arg(iz)?
Arg(z) +0.5π
What happens to the determinant if you swap two columns?
The sign is reversed.
How does a cyclic interchange affect the determinant?
It doesn’t
If a matrix has two identical columns, what is the determinant?
0
If you multiply one column of a matrix by K, what happens to the determinant?
Multiplied by K
If you multiply every column by a constant the determinant is multiplied by all the constants.
What is adj(M)?
The inverse of the matrix but with out being divided by the determinant
What is a cheaf ?
Line where three planes meet
What is the relationship between eigenvectors and eigenvalues.
(M^n)* vector =(value^n)*vector
How do you find eigen values?
Find the solutions to the characteristic equation.
How do you find eigen vectors from a characteristic equation?
Subtract the eigen value from the leading diagonal
Multiply be x,y,z
Rearrange the equations.
How do you find S and ¥ such that S¥S^-1 =M
Find eigen values for M Find corresponding vectors Put the vectors together for S Put values in diagonal for ¥ Make sure vectors and values are in same order.
If you have to find M to a power but know ¥ and S, what do you do?
Just put ¥ to that power and multiply by S and S^-1
How is the determinant of a matrix found from the characteristic equation?
It is the negative of the constant at the end provided the x³ term is positive.
What substitution would you use to integrate 1÷ (a+bx²)
X(b)^0.5 = tan(theta)
What substition would you use to integrate (a² -x²)^0.5
X=asin(theta)
What substition would you use to integrate (a²+x²)^0.5?
x =a sinh
What substition would you use to integrate (x²-a²)^0.5
X= a*cosh
What substitution would you use to intergrate (x²+1)^0.5
U= sinh(x)
What would you substitute intergrate (a²-x²)^-1.5 ?
X=2sin theta
X=2cos theta
